ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
2
fVIi.kh
?kkrkad rFkk dj.kh
fiNys ikB esa vkius nks ;k nks ls vf/kd okLrfod la[;kvksa dh xq.kk ds ckjs esa lh[kk gSA vki
fuEu dks cM+h vklkuh ls fy[k ldrs gSAa
4 × 4 × 4 = 64,11 × 11 × 11 × 11 = 14641 rFkk
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
vki ml leL;k ds fo"k; esa lksfp, tc 13 dks 15 ckj xq.kk fd;k tkuk gSA bls
13 × 13 × 13 ×.................15 ckj
:i esa fy[kuk dfBu gSA
ge bl dfBukbZ dks ?kkrkad ladsru ls ifjp; djkdj lqy>k;sx
a sA bl ikB esa ge ?kkrkad
dk vFkZ Li"V djsx
a s] ?kkrkadksa ds fu;eksa dks fy[kdj fl) djsx
a s rFkk mu fu;eksa dk iz;ksx
djuk lh[ksx
a sA ge la;qDr la[;kvksa dks vHkkT; la[;kvksa ds xq.kuQy ds :i esa izdV djsx
a sA
bl ikB ds vxys Hkkx esa ge la[;k a1/q dks a ds q osa ewy dk uke nsx
a sA ge vkidk ifjp;
dj.kh] dj.kh /kkr ls djk,sx
a sA ge dj.kh ds fu;eksa ds ckjs eas Hkh i<+x
sa s rFkk ,d dj.kh dk
ljyre :i Hkh Kkr djsx
a sA ge in ^^ifjes;dkjh xq.kd** dk vFkZ Hkh lh[ksx
a s rFkk nh xbZ
djf.k;ksa ds gj dk ifjes;dj.k djsx
a sA
mís';
bl ikB dks i<+us ds i'pkr vki leFkZ gks tk,sx
a s fd%
40
•
ckj&ckj vkus okys xq.ku dks ?kkrkadh; ladsru esa fy[k ldsx
a s rFkk mldk foykse fy[k
ldsx
a s(
•
?kkrkadh; ladsru esa fy[kh la[;kvksa ds vk/kkj rFkk ?kkrkad dh igpku dj ldsx
a s(
•
izkÑr la[;kvksa dks vHkkT; la[;kvksa ds vf}rh; xq.kuQy ds :i esa fy[k ldsx
a s(
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
•
?kkrkadksa ds fu;eksa dks fy[k ldsx
a s(
•
q
a a rFkk a ds vFkZ dh O;k[;k dj ldsx
a s(
•
?kkrkadksa ds fu;eksa dk iz;ksx dj ?kkrkadksa okys O;atdksa dks ljy dj ldasxs(
•
vifjes; la[;kvksa ds lewg esa ls djf.k;ksa dks igpku ldsx
a s(
•
dj.kh dk dj.kh?kkr rFkk dj.khxr Kkr dj ldsx
a s(
•
djf.k;ksa ds fu;e fy[k ldsx
a s(
•
,d nh xbZ dj.kh dks mlds ljyre :i esa fy[k ldasxs(
•
ltkrh; rFkk fotkrh; djf.k;ksa dk oxhZdj.k dj ldasxs(
•
fHkUu dj.kh?kkr dh djf.k;ksa dks ,d gh dj.kh?kkr dh djf.k;ksa esa cny ldsx
a s(
•
djf.k;kas ij pkj ewyHkwr lafØ;k,a dj ldsx
a s(
•
nh xbZ djf.k;ksa dks muds eku ds vuqlkj vkjksgh@vojksgh Øe esa fy[k ldsx
a s(
•
,d nh xbZ dj.kh dk ifjes;dkjh xq.kd Kkr dj ldsx
a s(
•
1
rFkk
a+b x
p
0,
–m
fVIi.kh
1
x + y , izdkj dh djf.k;ksa ds gj dk ifjes;dj.k dj ldasxs tgk¡
x rFkk y izkÑr la[;k,a gSa rFkk a vkSj b iw.kk±d gSa(
•
dj.kh okys O;atdksa dks ljy dj ldasxsA
visf{kr iwoZ Kku
•
vHkkT; la[;k,a
•
la[;kvksa ij pkj ewyHkwr lafØ;k,a
•
ifjes; la[;k,a
•
la[;kvksa esa Øe laca/k
2-1 ?kkrkadh; ladsru
fuEu xq.kuQyksa ij fopkj dhft,%
(i) 7 × 7
(ii) 3 × 3 × 3
(iii) 6 × 6 × 6 × 6 × 6
(i) esa la[;k 7 dks nks ckj xq.kk fd;k x;k gS] vr% 7 × 7 dks 72 ds :i esa fy[krs gSAa
(ii) esa la[;k 3 dks rhu ckj xq.kk fd;k x;k gS] vr% 3 × 3 × 3 dks 33 ds :i esa fy[krs gSAa
xf.kr
41
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
(iii) esa la[;k 6 dks ikap ckj xq.kk fd;k x;k gS] vr% 6 × 6 × 6 × 6 × 6 dks 65 ds :i esa fy[krs gSAa
72 dks “7 dh ?kkr 2” ;k “7 dh nwljh ?kkr” i<+rs gSA ;gk¡, 7 vk/kkj dgykrk gS vkSj 2 dks
?kkrkad ¼;k ?kkr½ dgrs gSAa
fVIi.kh
blh izdkj, 33 dks “3 dh ?kkr 3” ;k “3 dh rhljh ?kkr” i<+rs gSA ;gk¡, 3 vk/kkj gS rFkk 3 dks
?kkrkad ¼;k ?kkr½ dgrs gSAa
blh izdkj, 65 dks “6 dh ?kkr 5”;k “6 dh ikapoh ?kkr” i<+rs gSA ;gk¡, 6 vk/kkj gS rFkk 5 dks
?kkrkad ¼;k ?kkr½ dgrs gSAa
mijksDr ls ge dg ldrs gSa fd
fdlh la[;k dks Lo;a ls dbZ ckj xq.kk djus dk ladsru ?kkrkadh; ladsru vFkok ?kkrkadh;
:i dgykrk gSA
vr%, 5 × 5 × .... 20 ckj = 520 rFkk (–7) × (–7) × .... 10 ckj = (–7)10
520 esa 5 vk/kkj gS rFkk 20 ?kkrkad gSA
(–7)10 esa vk/kkj –7 rFkk ?kkrkad 10 gSA
blh izdkj] ?kkrkadh; ladsru dk iz;ksx dj ifjes; la[;k dks Lo;a ls ckj&ckj xq.kk djus
esa iz;ksx fd;k tkrk gSA
16
vr%,
3 3
⎛3⎞
× × .........1 6 ckj = ⎜ ⎟
5 5
⎝5⎠
rFkk
⎛ 1⎞ ⎛ 1⎞
⎛ 1⎞
⎜ − ⎟ × ⎜ − ⎟ × .......... 10 ckj = ⎜ – ⎟
⎝ 3⎠ ⎝ 3⎠
⎝ 3⎠
10
O;kid :i ls] ;fn a ,d ifjes; la[;k gS] tks Lo;a ls m ckj xq.kk dh tkrh gS] rks mls am
}kjk fy[kk tkrk gSA
;gk¡ a dks vk/kkj dgk tkrk gS rFkk m dks ?kkrkad ¼;k ?kkr½A
mijksDr ppkZ dks Li"V djus ds fy, vkb, dqN mnkgj.k yasA
mnkgj.k 2.1: fuEu eas ls izR;sd dks ljy dhft,%
(i ) ⎛⎜ 2 ⎞⎟
3
⎛ 3⎞
(ii) ⎜ − ⎟
⎝ 5⎠
⎝7⎠
8
⎛ 2 ⎞ 2 2 2 (2)
⎜ ⎟ = × × = 3=
⎝ 7 ⎠ 7 7 7 (7 ) 343
3
gy:
42
(i)
4
3
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
81
⎛ 3 ⎞ ⎛ 3 ⎞⎛ 3 ⎞⎛ 3 ⎞⎛ 3 ⎞ (− 3)
=
⎜ − ⎟ = ⎜ − ⎟⎜ − ⎟⎜ − ⎟⎜ − ⎟ =
4
625
⎝ 5 ⎠ ⎝ 5 ⎠⎝ 5 ⎠⎝ 5 ⎠⎝ 5 ⎠ (5)
4
(ii)
4
mnkgj.k 2.2: fuEu xq.ku dks ?kkrkadh; ladsru ds :i esa fyf[k,%
(i) (–5) × (–5) × (–5) × (–5) × (–5) × (–5) × (–5)
fVIi.kh
⎛3⎞ ⎛3⎞ ⎛3⎞ ⎛3⎞
(ii) ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟
⎝ 11 ⎠ ⎝ 11 ⎠ ⎝ 11 ⎠ ⎝ 11 ⎠
gy :
(i) (–5) × (–5) × (–5) × (–5) × (–5) × (–5) × (–5) = (–5)7
⎛3⎞ ⎛3⎞ ⎛3⎞ ⎛3⎞ ⎛3⎞
(ii) ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟ = ⎜ ⎟
⎝ 11 ⎠ ⎝ 11 ⎠ ⎝ 11 ⎠ ⎝ 11 ⎠ ⎝ 11 ⎠
4
mnkgj.k 2.3: fuEu esa ls izR;sd dks ?kkrkadh; ladsru esa O;Dr dhft, rFkk izR;sd esa
vk/kkj rFkk ?kkrkad fyf[k,%
(i) 4096
gy:
(ii)
125
729
(iii) – 512
(i) 4096 = 4 × 4 × 4 × 4 × 4 × 4
= (4)6
fodYir% 4096 = (2)12
vk/kkj = 2, ?kkrkad =12
;gk¡] vk/kkj = 4 vkSj ?kkr = 6
125
5 5 5 ⎛5⎞
(ii)
= × × =⎜ ⎟
729 9 9 9 ⎝ 9 ⎠
3
⎛5⎞
;gk¡] vk/kkj = ⎜ 9 ⎟ vkSj ?kkrkad = 3
⎝ ⎠
(iii) 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 29
;gk¡] vk/kkj = 2 vkSj ?kkrkad = 9
mnkgj.k 2.4: fuEu dks ljy dhft,
3
⎛3⎞ ⎛4⎞
⎜ ⎟ ×⎜ ⎟
⎝2⎠ ⎝3⎠
4
3
gy:
xf.kr
3 3 3 33
⎛3⎞
⎜ ⎟ = × × = 3
2 2 2 2
⎝2⎠
43
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
4
44
⎛4⎞
=
⎜ ⎟
34
⎝3⎠
3
4
33 4 4
33 28
⎛3⎞ ⎛4⎞
×
⎜ ⎟ ⎜ ⎟ = 3× 4 = 3× 4
2 3
2 3
⎝2⎠ ⎝3⎠
fVIi.kh
=
2 5 32
=
31
3
mnkgj.k 2.5: fuEu esa ls izR;sd dk foykse fyf[k, rFkk mUgsa ?kkrkadh; :i esa Hkh fyf[k,A%
⎛3⎞
(ii) ⎜ ⎟
⎝4⎠
5
(i) 3
gy:
(i)
2
⎛ 5⎞
(iii) ⎜ − ⎟
⎝ 6⎠
9
35 = 3 × 3 × 3 × 3 × 3
= 243
1
⎛1⎞
=⎜ ⎟
∴ 3 dk foykse =
243 ⎝ 3 ⎠
5
5
2
(ii)
32
⎛3⎞
⎜ ⎟ = 2
4
⎝4⎠
2
42
⎛3⎞
⎛4⎞
⎜
⎟
∴
dk foykse = 2 = ⎜ ⎟
3
⎝4⎠
⎝3⎠
2
⎛ 5⎞
(− 5)9
⎜− ⎟ =
69
⎝ 6⎠
9
(iii)
9
− 69 ⎛ − 6 ⎞
⎛ 5⎞
⎟
∴ ⎜ − ⎟ dk foykse = 9 = ⎜
5
⎝ 6⎠
⎝ 5 ⎠
9
p
mijksä mnkgj.kksa ls ge dg ldrs gSa fd ;fn q ,d 'kwU;srj ifjes; la[;k gS rFkk m ,d
m
m
⎛ p⎞
⎛q⎞
/kukRed iw.kk±d gS] rks ⎜⎜ ⎟⎟ dk foykse ⎜⎜ ⎟⎟ gSA
⎝q⎠
⎝ p⎠
44
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
ns[ksa vkius fdruk lh[kk 2-1
1. fuEu dks ?kkrkadh; :i esa fyf[k,%
fVIi.kh
(i) (–7) × (–7) × (–7) × (–7)
⎛3⎞ ⎛3⎞
(ii) ⎜ ⎟ × ⎜ ⎟ × .... 10 ckj
⎝4⎠ ⎝4⎠
⎛ 5⎞ ⎛ 5⎞
(iii) ⎜ − ⎟ × ⎜ − ⎟ × .... 20 ckj
⎝ 7⎠ ⎝ 7⎠
2. fuEu esa ls izR;sd dk vk/kkj rFkk ?kkrkad fyf[k,%
5
⎛ 2⎞
(iii) ⎜ − ⎟
⎝ 11 ⎠
4
(i) (–3)
(ii) (7)
8
3. fuEu esa ls izR;sd dks ljy dhft,%
⎛3⎞
(i) ⎜ ⎟
⎝7⎠
4
⎛ –2⎞
(ii) ⎜
⎟
⎝ 9 ⎠
4
⎛ 3⎞
(iii) ⎜ – ⎟
⎝ 4⎠
3
4. fuEu esa ls izR;sd dks ljy dhft,%
5
⎛7⎞ ⎛3⎞
(i) ⎜ ⎟ × ⎜ ⎟
⎝3⎠ ⎝7⎠
2
6
⎛ 5⎞ ⎛ 3⎞
(ii) ⎜ − ⎟ ÷ ⎜ − ⎟
⎝ 6⎠ ⎝ 5⎠
2
5. fuEu esa ls izR;sd dk foykse Kkr dhft,%
(i) 35
(ii) (–7)4
⎛ 3⎞
(iii) ⎜ − ⎟
⎝ 5⎠
4
2-2 vHkkT; xq.ku[kaM
Lej.k dhft, fd dksbZ la;qä la[;k vHkkT; la[;kvksa ds xq.kuQyksa ds :i esa fy[kh tk
ldrh gSA vkb, ge la;qä la[;k,a 72] 760 rFkk 7623 ysaA
xf.kr
45
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
(i) 72 = 2 × 2 × 2 × 3 × 3
= 2 3 × 32
(ii) 760 = 2 × 2 × 2 × 5 × 19
2
2
2
3
72
36
18
9
3
3
3
7
11
7623
2541
847
121
11
= 23 × 51 ×191
fVIi.kh
(iii) 7623 = 3 × 3 × 7 × 11 ×11
= 32 × 71 × 112
2
2
2
5
760
380
190
95
19
ge ns[k ldrs gSa fd 1 ds vfrfjDr izR;sd izkÑr la[;k] vf}rh; :i ls vHkkT; la[;kvksa
dh ?kkrksa ds xq.kuQy ds :i esa fy[kh tk ldrh gS] ;fn muds vkus okys Øe dks NksM+ fn;k
tk,A vkb, dqN mnkgj.k ysAa
mnkgj.k 2.6: 24300 dks vHkkT; la[;kvksa ds ?kkrkadh; :i esa O;ä dhft,%
gy: 24300 = 3 × 3 × 3 × 3 × 2 × 2 × 5 × 5 × 3
∴ 24300 = 22 × 35 ×52
mnkgj.k 2.7: 98784 dks vHkkT; la[;kvksa ds
?kkrkadh; :i esa O;ä dhft,%
gy:
2
2
2
2
2
3
3
7
7
98784
49392
24696
12348
6174
3087
1029
343
49
7
2
2
3
3
3
3
3
5
24300
12150
6075
2025
675
225
75
25
5
∴ 98784 = 25 × 32 × 73
ns[ksa vkius fdruk lh[kk 2-2
1. fuEufyf[kr esa ls izR;sd dks vHkkT; la[;kvksa ds ?kkrkdksa ds xq.kuQy ds :i esa O;Dr
dhft,%
(i) 429
46
(ii) 648
(iii) 1512
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
2. fuEu esa ls izR;sd dks vHkkT; la[;kvksa ds ?kkrkadh; :i esa O;ä dhft,%
(i) 729
(iv)
(ii) 512
1331
4096
(v) −
(iii) 2592
243
32
fVIi.kh
2-3 ?kkrkadksa ds fu;e
fuEu ij fopkj dhft,%
(i) 32 × 33
= (3 × 3) × (3 × 3 × 3) = (3 × 3 × 3 × 3 × 3)
= 35 = 32 + 3
(ii) (–7)2 × (–7)4 = [(–7) × (–7)] × [(–7) × (–7) × (–7) × (–7)]
= [ (–7) × (–7) × (–7) × (–7) × (–7) × (–7)]
= (–7)6 = (–7)2+4
3
4
⎛3⎞ ⎛3⎞ ⎛3 3 3⎞ ⎛3 3 3 3⎞
(iii) ⎜ ⎟ × ⎜ ⎟ = ⎜ × × ⎟ × ⎜ × × × ⎟
⎝4⎠ ⎝4⎠ ⎝4 4 4⎠ ⎝4 4 4 4⎠
⎛3 3 3 3 3 3 3⎞
=⎜ × × × × × × ⎟
⎝4 4 4 4 4 4 4⎠
7
⎛3⎞ ⎛3⎞
= ⎜ ⎟ =⎜ ⎟
⎝4⎠ ⎝4⎠
3+ 4
(iv) a3 × a4 = (a × a × a) × (a × a × a × a) = a7 = a3+4
mijksä mnkgj.kksa ls ge voyksdu djrs gSa fd
fu;e 1: ;fn a ,d 'kwU;srj ifjes; la[;k gS rFkk m rFkk n nks /ku iw.kk±d gS]a rks
am × an = am+n
3
5
⎛ 3⎞ ⎛ 3⎞
mnkgj.k 2.8: ⎜ − ⎟ × ⎜ − ⎟ dk eku Kkr dhft,A
⎝ 2⎠ ⎝ 2⎠
gy:
3
2
;gk¡ a = − , m = 3 and n = 5.
3
5
⎛ 3⎞ ⎛ 3⎞ ⎛ 3⎞
∴ ⎜− ⎟ ×⎜− ⎟ = ⎜− ⎟
⎝ 2⎠ ⎝ 2⎠ ⎝ 2⎠
xf.kr
3+ 5
8
⎛ 3 ⎞ 6561
= ⎜− ⎟ =
256
⎝ 2⎠
47
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
mnkgj.k 2.9: fuEu dk eku Kkr dhft,%
2
⎛7⎞ ⎛7⎞
⎜ ⎟ ×⎜ ⎟
⎝ 4⎠ ⎝ 4⎠
fVIi.kh
gy:
3
igys dh rjg
2
3
⎛7⎞ ⎛7⎞ ⎛7⎞
⎜ ⎟ ×⎜ ⎟ = ⎜ ⎟
⎝ 4⎠ ⎝ 4⎠ ⎝ 4⎠
2 +3
5
⎛ 7 ⎞ 16807
=⎜ ⎟ =
1024
⎝4⎠
vc fuEu dks if<+,%
(i) 75 ÷ 73 =
75 7 × 7 × 7 × 7 × 7
=
= 7 × 7 = 7 2 = 7 5 −3
3
7
7×7×7
(ii) (–3)7 ÷ (–3)4 =
(− 3)7 = (− 3)× (− 3)× (− 3)× (− 3)× (− 3)× (− 3)× (− 3)
(− 3)× (− 3)× (− 3)× (− 3)
(− 3)4
= (− 3)(− 3)(− 3) = (− 3)3 = (–3)7– 4
mijksDr esa ge ns[krs gSa fd
fu;e 2: ;fn a ,d 'kwU;srj ifjes; la[;k gS] rFkk m rFkk n /ku iw.kk±d gSa (m > n), rks
am ÷ an = am–n
16
13
⎛ 35 ⎞
⎛ 35 ⎞
mnkgj.k 2.10: ⎜ ⎟ ÷ ⎜ ⎟
⎝ 25 ⎠
⎝ 25 ⎠
16
dk eku Kkr dhft,A
13
⎛ 35 ⎞
⎛ 35 ⎞
⎜ ⎟ ÷⎜ ⎟
⎝ 25 ⎠
⎝ 25 ⎠
gy:
⎛ 35 ⎞
=⎜ ⎟
⎝ 25 ⎠
16 −13
3
3
343
⎛ 35 ⎞
⎛7⎞
=⎜ ⎟ =⎜ ⎟ =
125
⎝ 25 ⎠
⎝5⎠
fu;e 2 esa m < n ⇒ n > m,
rks
a m ÷ a n = a −(n − m ) =
1
a
m− n
fu;e 3: tc n > m gS
am ÷ an =
48
1
a
m−n
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
6
9
⎛3⎞ ⎛3⎞
mnkgj.k 2.11: ⎜ ⎟ ÷ ⎜ ⎟ dk eku Kkr dhft,A
⎝7⎠ ⎝7⎠
gy:
3
, m = 6 rFkk n = 9.
7
;gk¡ a =
6
fVIi.kh
1
9
⎛ 3 ⎞ ⎛ 3 ⎞ ⎛ 3 ⎞ 9 −6
∴ ⎜ ⎟ ÷⎜ ⎟ = ⎜ ⎟
⎝7⎠ ⎝7⎠ ⎝7⎠
=
7 3 343
=
33 27
vkb, fuEu dks ns[ksa
(i)
(3 )
3 2
= 33 × 33 = 33+3 = 36 = 33×2
5
⎡⎛ 3 ⎞ 2 ⎤ ⎛ 3 ⎞ 2 ⎛ 3 ⎞ 2 ⎛ 3 ⎞ 2 ⎛ 3 ⎞ 2 ⎛ 3 ⎞ 2
(ii) ⎢⎜ ⎟ ⎥ = ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟
⎢⎣⎝ 7 ⎠ ⎥⎦ ⎝ 7 ⎠ ⎝ 7 ⎠ ⎝ 7 ⎠ ⎝ 7 ⎠ ⎝ 7 ⎠
⎛3⎞
⎜ ⎟
⎝7⎠
2 + 2+ 2 + 2 + 2
10
⎛3⎞
⎛3⎞
=⎜ ⎟ =⎜ ⎟
⎝7⎠
⎝7⎠
2×5
mijksDr nks fLFkfr;ksa ls ge fuEu ifj.kke fudkyrs gS%a
fu;e 4: ;fn a ,d dksbZ 'kwU;srj ifjes; la[;k gS rFkk m rFkk n nks /ku iw.kk±d gS]a rks
(a )
m n
= a mn
vkb, ge ,d mnkgj.k ysAa
3
⎡⎛ 2 ⎞ 2 ⎤
mnkgj.k 2.12: ⎢⎜ 5 ⎟ ⎥ dk eku Kkr dhft,A
⎣⎢⎝ ⎠ ⎦⎥
3
gy:
⎡⎛ 2 ⎞ 2 ⎤ ⎡ 2 ⎤ 2×3 ⎛ 2 ⎞6
64
⎢⎜ ⎟ ⎥ = ⎢ ⎥ = ⎜ ⎟ =
⎝ 5 ⎠ 15625
⎣⎢⎝ 5 ⎠ ⎦⎥ ⎣ 5 ⎦
2.3.1 'kwU; ?kkrkad
Lej.k dhft, fd a m ÷ a n = a m − n , ;fn m > n
xf.kr
49
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
=
1
a
n− m
, ;fn n > m
vkb, ge og voLFkk ysa tc m = n
∴ a m ÷ a m = a m−m
fVIi.kh
am
= a0
am
⇒ 1 = a0
⇒
fu;e 5: ;fn a dksbZ 'kwU;srj ifjes; la[;k gS] rks ao = 1.
mnkgj.k 2.13: eku Kkr dhft,
⎛2⎞
(i) ⎜ ⎟
⎝7⎠
0
⎛ −3⎞
⎟
(ii) ⎜
⎝ 4 ⎠
0
0
⎛2⎞
(i) a = 1 dk iz;ksx djds gesa feyrk gS ⎜ ⎟ = 1
⎝7⎠
gy:
0
0
⎛ −3⎞
⎟ = 1.
(ii) fQj a0 = 1 dk iz;ksx djds gesa feyrk gS ⎜
⎝ 4 ⎠
ns[ksa vkius fdruk lh[kk 2-3
1. fuEu dks ljy djds ifj.kke dks ?kkrkadh; :i esa O;Dr dhft,%
3
⎛3⎞ ⎛3⎞
(ii) ⎜ ⎟ × ⎜ ⎟
⎝4⎠ ⎝4⎠
(i) (7)2 ×(7)3
2
1
2
⎛ 7⎞ ⎛ 7⎞ ⎛ 7⎞
(iii) ⎜ − ⎟ × ⎜ − ⎟ × ⎜ − ⎟
⎝ 8⎠ ⎝ 8⎠ ⎝ 8⎠
3
2. fuEu dks ljy djds ifj.kke dks ?kkrkadh; :i esa O;Dr dhft,%
8
(i) (− 7 ) ÷ (− 7 )
9
7
⎛3⎞ ⎛3⎞
(ii) ⎜ ⎟ ÷ ⎜ ⎟
⎝4⎠ ⎝4⎠
2
18
⎛−7⎞
⎛ −7⎞
⎟ ÷⎜
⎟
(iii) ⎜
⎝ 3 ⎠
⎝ 3 ⎠
3
3. fuEu dks ljy djds ifj.kke dks ?kkrkadh; :i esa O;Dr dhft,%
( )
(i) 2 6
50
3
⎡⎛ 3 ⎞3 ⎤
(ii) ⎢⎜ ⎟ ⎥
⎢⎣⎝ 4 ⎠ ⎥⎦
2
⎡⎛ 5 ⎞3 ⎤
(iii) ⎢⎜ − ⎟ ⎥
⎢⎣⎝ 9 ⎠ ⎥⎦
5
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
5
0
0
⎛ 11 ⎞ ⎛ 15 ⎞
⎛ 7⎞ ⎛ 7⎞
(iv) ⎜ ⎟ × ⎜ ⎟ (v) ⎜ − ⎟ × ⎜ − ⎟
⎝3⎠ ⎝7⎠
⎝ 11 ⎠ ⎝ 11 ⎠
3
4. fuEu esa ls dkSu&dkSu ls dFku lR; gS\a
fVIi.kh
5
(i) 73 × 73 = 76
4
7
2
⎡⎛ 4 ⎞5 ⎤ ⎛ 4 ⎞9
(iii) ⎢⎜ ⎟ ⎥ = ⎜ ⎟
⎢⎣⎝ 9 ⎠ ⎥⎦ ⎝ 9 ⎠
⎡⎛ 3 ⎞6 ⎤ ⎛ 3 ⎞8
(iv) ⎢⎜ ⎟ ⎥ = ⎜ ⎟
⎢⎣⎝ 19 ⎠ ⎥⎦ ⎝ 19 ⎠
0
2
⎛3⎞
(v) ⎜ ⎟ = 0
⎝ 11 ⎠
5
2
⎛3⎞ ⎛3⎞ ⎛3⎞
(ii) ⎜ ⎟ × ⎜ ⎟ = ⎜ ⎟
⎝ 11 ⎠ ⎝ 11 ⎠ ⎝ 11 ⎠
9
⎛ 3⎞
(vi) ⎜ − ⎟ = −
4
⎝ 2⎠
0
⎛ 8 ⎞ ⎛7⎞ ⎛ 8 ⎞
(vii) ⎜ ⎟ × ⎜ ⎟ = ⎜ ⎟
⎝ 15 ⎠ ⎝ 6 ⎠ ⎝ 15 ⎠
5
2-4 _.kkRed iw.kk±d ?kkrkad ds :i esa
i)
ge tkurs gSa fd 5 dk O;qRØe
1
gSA ge bls '5–1' fy[krs gSa rFkk '5 dh ?kkr –1' i<+rs
5
gSaA
ii) (–7) dk O;qRØe −
iii) 52 dk O;qRØe
1
gSA ge bls '(–7)–1' fy[krs gSa rFkk '(–7) dh ?kkr –1' i<+rs gSaA
7
1
gSA ge bls '5–2' fy[krs gSa rFkk ‘5 dh ?kkr (–2)’ i<+rs gSAa
52
mijksDr ls ge fuEu ifj.kke izkIr djrs gSAa
;fn a ,d 'kwU;srj ifjes; la[;k gS rFkk m ,d /kuiw.kk±d gS] rks am dk O;qRØe
1
gksrk gSA bls 'a–m' fy[krs gSa rFkk ‘a dh ?kkr (–m)’ i<+rs gSaA vr%
am
1
= a −m
am
vkb, ,d mnkgj.k ysAa
xf.kr
51
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
mnkgj.k 2.14: fuEu esa ls izR;sd dks /kukRed ?kkrkadksa ds lkFk fyf[k,%
⎛3⎞
(i) ⎜ ⎟
⎝8⎠
fVIi.kh
−2
⎛ 4⎞
(ii) ⎜ − ⎟
⎝ 7⎠
−7
gy:
−2
1
1 82 ⎛ 8 ⎞
⎛3⎞
(i) ⎜ ⎟ =
= 2 = 2 =⎜ ⎟
2
3
3 ⎝3⎠
⎝8⎠
⎛3⎞
⎜ ⎟
2
8
⎝8⎠
2
−7
1
77
⎛ 4⎞
⎛ 7⎞
(ii) ⎜ − ⎟ =
=
= ⎜− ⎟
7
7
(− 4) ⎝ 4 ⎠
⎝ 7⎠
⎛ 4⎞
⎜− ⎟
⎝ 7⎠
7
mijksDr mnkgj.k ls ge fuEu ifj.kke izkIr djrs gSa
p
⎛ p⎞
;fn q ,d 'kwU;srj ifjes; la[;k gS rFkk m dksbZ /kuiw.kk±d gS] rks ⎜⎜ ⎟⎟
⎝q⎠
−m
m
qm ⎛ q ⎞
= m = ⎜⎜ ⎟⎟ .
p
⎝ p⎠
2-5 iw.kk±dh; ?kkrkadksa ds fy, ?kkrkadksa ds fu;e
_.kkRed iw.kk±dksa dk ?kkrkadksa ds :i esa vFkZ fuf'pr djus ds ckn vkb, ns[ksa fd bl ikB
esa igys i<+s x, ?kkrkadksa ds fu;e D;k _.kkRed ?kkrkadksa ds fy, Hkh lR; gSAa mnkgj.kr;k
−4
3
3
1
⎛3⎞
⎛3⎞
⎛ 3⎞ ⎛ 3⎞
(i) ⎜ ⎟ × ⎜ ⎟ =
×⎜ ⎟ = ⎜ ⎟
4
⎝5⎠
⎝5⎠ ⎛3⎞ ⎝5⎠ ⎝5⎠
⎜ ⎟
⎝5⎠
−2
−3
−3
−7
3– 4
1
1
1
⎛ 2⎞
⎛ 2⎞
⎛ 2⎞
(ii) ⎜ − ⎟ × ⎜ − ⎟ =
×
=
= ⎜− ⎟
2
3
2 +3
⎝ 3⎠
⎝ 3⎠
⎝ 3⎠
⎛ 2⎞ ⎛ 2⎞
⎛ 2⎞
⎜− ⎟ ⎜− ⎟
⎜− ⎟
⎝ 3⎠ ⎝ 3⎠
⎝ 3⎠
⎛ 3⎞
⎛ 3⎞
(iii) ⎜ − ⎟ ÷ ⎜ − ⎟
⎝ 4⎠
⎝ 4⎠
3
7
− 2−3
⎛ 3⎞ ⎛ 3⎞
=
÷
=
×⎜− ⎟ = ⎜− ⎟
3
7
3
⎛ 3⎞ ⎛ 3⎞
⎛ 3⎞ ⎝ 4⎠ ⎝ 4⎠
⎜− ⎟ ⎜− ⎟
⎜− ⎟
⎝ 4⎠ ⎝ 4⎠
⎝ 4⎠
1
1
1
7 −3
3
⎛ ⎛ 2 ⎞ −2 ⎞ ⎡⎛ 7 ⎞ 2 ⎤ ⎛ 7 ⎞ 6 ⎛ 2 ⎞ −6 ⎛ 2 ⎞ − 2×3
(iv) ⎜ ⎜ ⎟ ⎟ = ⎢⎜ ⎟ ⎥ = ⎜ ⎟ = ⎜ ⎟ = ⎜ ⎟
⎜ ⎝ 7 ⎠ ⎟ ⎢⎝ 2 ⎠ ⎥ ⎝ 2 ⎠ ⎝ 7 ⎠
⎝7⎠
⎦
⎠ ⎣
⎝
52
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
mijksDr mnkgj.kksa ls ge ns[krs gSa fd fu;e 1 ls fu;e 5 rd lHkh fu;e _.kkRed
?kkrkadksa ds fy;s Hkh lR; gSaA
;fn a rFkk b dksbZ 'kwU;srj ifjes; la[;k,¡ gSa rFkk m, n dksbZ iw.kk±d gS] rks
1. am × an
= am+n
2. am ÷ an
= am–n ;fn m > n
fVIi.kh
= a–(n–m) =
1
a
;fn n > m
n− m
3. (am)n = amn
4. (a × b)m = am × bm
ns[ksa vkius fdruk lh[kk 2-4
−2
p
⎛ −3⎞
⎟ dks
1. ⎜
ds :i dh ifjes; la[;k ds :i esa O;Dr dhft,A
q
⎝ 7 ⎠
2. fuEu dks ,d ifjes; la[;k ds /kukRed ?kkrkad ds :i esa O;Dr dhft,%
⎛ 3⎞
(i) ⎜ ⎟
⎝7⎠
⎡⎛ 3 ⎞ −3 ⎤
(iii) ⎢⎜ ⎟ ⎥
⎣⎢⎝ 13 ⎠ ⎦⎥
−4
(ii) 125 × 12 −3
4
3. fuEu dks ,d ifjes; la[;k ds _.kkRed ?kkrkad ds :i esa O;Dr dhft,%
⎛3⎞
(i) ⎜ ⎟
⎝7⎠
4
⎡⎛ 3 ⎞ 2 ⎤
(iii) ⎢⎜ − ⎟ ⎥
⎣⎢⎝ 4 ⎠ ⎦⎥
[ ]
(ii) (7 )2
5
5
4. ljy dhft,%
−3
⎛3⎞
⎛3⎞
(i) ⎜ ⎟ × ⎜ ⎟
⎝2⎠
⎝2⎠
−3
7
⎛ 2⎞
⎛ 2⎞
(ii) ⎜ − ⎟ × ⎜ − ⎟
⎝ 3⎠
⎝ 3⎠
−4
4
⎛ 7⎞
⎛ 7⎞
(iii) ⎜ − ⎟ ÷ ⎜ − ⎟
⎝ 5⎠
⎝ 5⎠
−7
5. fuEu esa ls dkSu&dkSu ls dFku lR; gS\a
(i) a–m × an = a–m–n
(ii) (a–m)n = a–mn
(iii) a × b = (ab)
m
m
m
⎛a⎞
(iv) a ÷ b = ⎜ ⎟
⎝b⎠
m
m
m
(v) a–m × ao = am
xf.kr
53
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
2-6 ap/q ls vfHkizk;
geus ns[kk gS fd m rFkk n ds lHkh iw.kk±dh; ekuksa ds fy,
am × an = am+n
fVIi.kh
;fn a ,d /kukRed ifjes; la[;k gS rFkk q ,d izkÑr la[;k gS rks ge a1/q dks fdl izdkj
ls ifjHkkf"kr djrs gSAa
fuEu xq.ku ij fopkj dhft,%
1
q
1
q
1
q
1
q
a × a × a ........× a = a
1 1 1
+ + + .....q ckj
q q q
q ckj
q
= aq = a
⎛ 1q ⎞
nwljs 'kCnksa eas ⎜⎜ a ⎟⎟ dh ?kkr q = a gSA
⎝ ⎠
1
nwljs 'kCnksa esa a q , a dk qoka ewy gS gS rFkk bls
q
a ds :i esa fy[krs gSaA
mnkgj.kr;k
1
4
1
4
1
4
1
4
7 ×7 ×7 ×7 = 7
1 1 1 1
+ + +
4 4 4 4
4
4
= 7 = 71 = 7
vkb, a dh ifjes; ?kkr dks ifjHkkf"kr djsAa
;fn a ,d /kukRed la[;k gS] p ,d iw.kk±d la[;k gS rFkk q ,d izkÑr la[;k gS] rks
p
q
aq = ap
ge ns[krs gSa fd
p
q
p
q
p
q
p
q
a × a × a ........ × a = a
p p p
+ + + .....q ckj
q q q
=a
p
.q
q
= ap
q ckj
p
q
q
∴a = a p
vr% ap/q, ap dk qoka ewy gSA
2
vr% 7 3 , 72 dk ?kuewy gSA
vkb, vc ifjes; ?kkrkadksa ds fy, ?kkrkadksa ds fu;e fy[ks%a
54
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
(i) am × an = am+n
(ii) am ÷ an = am–n
(iii) (am)n = amn
(iv) (ab)m = am bm
fVIi.kh
m
am
⎛a⎞
(v) ⎜ ⎟ = m
b
⎝b⎠
mijksDr fu;eksa ds lR;kiu ds fy, vkb, dqN mnkgj.k ysAa
mnkgj.k 2.15: eku Kkr dhft,%
(i) (625)
1
4
(ii) (243)
⎛ 16 ⎞
(iii) ⎜ ⎟
⎝ 81 ⎠
2
5
−3 / 4
gy:
(i)
1
(625)4 = (5 × 5 × 5 × 5) 4 = (54 )4 = 5
1
1
2
2
( )
(ii) (243)5 = (3 × 3 × 3 × 3 × 3) 5 = 35
−3
2
5
4×
1
4
=5
5×
=3
2
5
= 32 = 9
−3
⎛ 16 ⎞ 4 ⎛ 2 × 2 × 2 × 2 ⎞ 4
(iii) ⎜ ⎟ = ⎜
⎟
⎝ 3× 3× 3× 3 ⎠
⎝ 81 ⎠
−3
⎛ −3 ⎞
⎟
4 ⎠
⎡⎛ 2 ⎞ 4 ⎤ 4 ⎛ 2 ⎞ 4×⎜⎝
= ⎢⎜ ⎟ ⎥ = ⎜ ⎟
⎝3⎠
⎣⎢⎝ 3 ⎠ ⎦⎥
⎛ 2⎞
=⎜ ⎟
⎝ 3⎠
−3
3
27
⎛ 3⎞
=⎜ ⎟ =
8
⎝ 2⎠
ns[ksa vkius fdruk lh[kk 2-5
1. fuEu esa ls izR;sd dks ljy dhft,%
(i) (16)
3
⎛ 27 ⎞
(ii) ⎜
⎟
⎝ 125 ⎠
4
−
2
3
2. fuEu esa ls izR;sd dks ljy dhft,%
1
1
(i) (625)− 4 ÷ (25)− 2
xf.kr
55
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
−
1
1
3
⎛ 7 ⎞ 4 ⎛ 7 ⎞2 ⎛ 7 ⎞4
(ii) ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟
⎝8⎠
⎝8⎠ ⎝8⎠
−
3
1
3
⎛ 13 ⎞ 4 ⎛ 13 ⎞ 4 ⎛ 13 ⎞ 2
(iii) ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟
⎝ 16 ⎠ ⎝ 16 ⎠
⎝ 16 ⎠
fVIi.kh
2-7 dj.kh
ge igys ikB esa lh[k pqds gSa fd 2 , 3 rFkk 5 izdkj dh la[;k,a vifjes; la[;k,a gSAa
vc ge ,d fo'ks"k izdkj dh vifjes; la[;kvksa dk v/;;u djsx
a s tks dj.kh dgykrh gSAa
,d dj.kh dks n x ds :i okyh ml /kukRed vifjes; la[;k ds :i esa ifjHkkf"kr fd;k
tkrk gS ftlesa x dk noka Bhd ewy Kkr djuk laHko u gks] tgk¡ x ,d /kukRed ifjes; la[;k gSA
la[;k
n
x ,d dj.kh gksxh ;fn vkSj dsoy ;fn
(i) ;g ,d vifjes; la[;k gksA
(ii) ;g ,d /kukRed ifjes; la[;k dk ewy gksA
2.7.1 dqN ifjHkk"kk,a
dj.kh n x , eas fpUg
dks dj.kh fpUg dgk tkrk gSA ?kkr ‘n’ dks ^dj.kh dh ?kkr* rFkk
x dks ^dj.khxr* dgrs gaSA
uksV: i)
ii)
tc dj.kh dh ?kkr ugha fy[kh gks rks mls 2 fy;k tkrk gSA mnkgj.kr;k]
7 (= 2 7 ) esa dj.kh dh ?kkr 2 gSA
3
8 dj.kh ugha gS D;ksfa d bldk Bhd eku 2 izkIr gks tkrk gS] tks ,d ifjes;
la[;k gSA
iii)
2 + 2 , gkykafd ,d vifjes; la[;k gS] ij dj.kh ugha gS D;ksfa d ;g ,d
vifjes; la[;k dk oxZey
w gSA
2-8 'kq) rFkk fefJr djf.k;ka
i)
56
,slh dj.kh ftldk ,d ifjes; xq.ku[kaM la[;k 1 gS rFkk nwljk xq.ku[akM ,d vifjes;
la[;k gS 'kq) dj.kh dgykrh gSA mnkgj.k ds fy, 5 16 vkSj 3 50 'kq) djf.k;ka gSAa
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
ii) ,slh dj.kh] ftldk ifjes; xq.ku[kaM 1 ds vfrfjDr vU; ifjes; la[;k gks tcfd vU;
xq.ku[kaM vifjes; la[;k gS] fefJr dj.kh dgykrh gSA mnkgj.k ds fy, 2 3 vkSj 3 3 7
fefJr djf.k;ka gSaA
fVIi.kh
2-9 dj.kh dh ?kkr
dj.kh 53 4 , esa 5 dks dj.kh dk xq.kkad] 3 dks dj.kh dh ?kkr rFkk 4 dks dj.khxr dgrs gSAa
vkb, dqN mnkgj.k ysAa
mnkgj.k 2.16: crkb, fuEu esa ls dkSu&dkSu lh djf.k;k¡ gS%a
(i) 49
gy:
(ii) 96
(i)
(iii) 3 81
(iv) 3 256
49 = 7, tks ,d ifjes; la[;k gSA
vr% 49 ,d dj.kh ugha gSA
96 = 4 × 4 × 6 = 4 6
(ii)
∴
96 ,d vifjes; la[;k gSA
⇒ 96 ,d dj.kh gSA
(iii)
3
81 = 3 3 × 3 × 3 × 3 = 33 3 , tks ,d vifjes; la[;k gSA
∴
(iv)
3
3
81 ,d dj.kh gSA
256 = 3 4 × 4 × 4 × 4 = 43 4
vr% 3 256 ,d vifjes; la[;k gSA
⇒
3
256 ,d dj.kh gSA
vr% (ii), (iii) rFkk (iv) dj.kh gSAa
mnkgj.k 2.17: fuEu esa ls izR;sd esa dj.kh dh ?kkr rFkk dj.khxr Kkr dhft,%
(i) 5 117
gy:
xf.kr
(ii) 162
(iii) 4 213
(iv) 4 214
(i)
dj.kh dh ?kkr 5 gS rFkk dj.khxr 117 gSA
(ii)
dj.kh dh ?kkr 2 gS rFkk dj.khxr 162 gSA
(iii)
dj.kh dh ?kkr 4 gS rFkk dj.khxr 213 gSA
(iv)
dj.kh dh ?kkr 4 gS rFkk dj.khxr 214 gSA
57
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
mnkgj.k 2.18: fuEu esa ls 'kq) rFkk fefJr djf.k;ka igpku dj fyf[k,%
(i) 42
(ii) 4 3 18
gy:
(i)
fVIi.kh
(iii) 2 4 98
42 ,d 'kq) dj.kh gSA
(ii) 4 3 18 ,d fefJr dj.kh gSA
(iii) 2 4 98 ,d fefJr dj.kh gSA
2-10 dj.kh fpUgksa ds fu;e
uhps djf.k;ksa ds dqN fu;e ¼miifr ds fcuk½ fn;s tk jgs gS%a
(i)
[ a] = a
(ii)
n
a n b = n ab
n
a n a
=
b
b
(iii)
n
n
n
tgk¡ a rFkk b /kukRed ifjes; la[;k,a gSa rFkk n ,d /ku iw.kk±d gSA
vkb, dqN mnkgj.k ysdj Li"V djsAa
mnkgj.k 2.19: fuEu esa ls dkSu&dkSu lh djf.k;k¡ gSa rFkk dkSu lh ugha gS\a dj.kh fpUgksa
ds fu;eksa dk iz;ksx dj fuf'pr dhft,A
gy:
(i) 5 × 80
(ii) 2 15 ÷ 4 10
(iii) 3 4 × 3 16
(iv) 32 ÷ 27
5 × 80 = 5 × 80 = 400 = 20
(i)
tks ,d ifjes; la[;k gSA
∴
(ii)
2 15 ÷ 4 10 =
=
58
5 × 80 ,d dj.kh ugha gSA
2 15
15
=
4 10 2 10
15
15
3
=
=
, tks vifjes; gSA
8
2 × 2 × 10
40
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
∴ 2 15 ÷ 4 10 ,d dj.kh gS
(iii)
(iv)
3
4 × 3 16 = 3 64 = 4, vr% ;g ,d dj.kh ugha gSA
32
32
=
, tks vifjes; gS
27
27
32 ÷ 27 =
∴
fVIi.kh
32 ÷ 27 ,d dj.kh gSA
ns[ksa vkius fdruk lh[kk 2-6
1. fuEu esa ls izR;sd ds fy, dj.kh dh ?kkr rFkk dj.khxr fyf[k,%
(ii) 6 343
(i) 4 64
(iii) 119
2. crkb, fuEu esa ls dkSu&dkSu lh djf.k;ak gS%a
(i) 3 64
(ii) 4 625
(iii) 6 216
(iv) 5 × 45
(v) 3 2 × 5 6
3. fuEu esa ls 'kq) rFkk fefJr djf.k;ak igpku dj fyf[k,%
(i) 32
(ii) 2 3 12
(iii) 13 3 91
(iv) 35
2-11 djf.k;ksa ds fu;e
Lej.k dhft, djf.k;ksa dks fHkUukRed ?kkrkadksa okyh la[;kvksa ds :i esa O;Dr fd;k tk
ldrk gSA blfy, bl ikB esa fn;s x, ?kkrkadksa ds fu;e djf.k;ksa ij Hkh ykxw gksrs gSAa vkb,
mUgsa iqu% Lej.k djsAa
1
(i)
n
n
(ii)
1
1
x . n y = n xy vFkok xn .y n = (xy)n
n
1
n
1
⎛ x ⎞n
x
x
x
vFkok 1 = ⎜⎜ ⎟⎟
=n
y
y
y
yn ⎝ ⎠
1
(iii)
m n
xf.kr
1
1
⎛ 1 ⎞m
⎛ 1 ⎞n
x = mn x = n m x vFkok ⎜⎜ x n ⎟⎟ = x mn = ⎜⎜ x m ⎟⎟
⎝ ⎠
⎝ ⎠
59
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
fVIi.kh
(iv)
n
(v)
m
m
n
( )
x = x vFkok x
m
p
x =
mn
x
pn
1
m n
vFkok (x
=x
)
1
p m
m
n
p
m
=x =x
pn
mn
( )
= x pn
1
mn
;gk¡ x rFkk y /kukRed ifjes; la[;k,a gSa rFkk m, n vkSj p /kuiw.kk±d gSAa
vkb, bu fu;eksa dks ge mnkgj.kksa }kjk Li"V djsAa
(i)
3
(ii)
(5)3
1
(9)3
3
3
1
(iii)
3 2
(iv)
5
1
3
1
3
1
5
⎛ 5 ⎞3
=⎜ ⎟ =3
9
⎝9⎠
⎛
7 = 7 = ⎜⎜ 7
⎝
1
2
3
( )
4 = 4
3
1
8 = 3 × 8 = (24)3 = 3 24 = 3 3 × 8
1
3 5
1
2
1
3
1
⎞
⎟ = 7 6 = 6 7 = 2×3 7 = 2
⎟
⎠
3
5
3
7
9
15
= 4 = 4 = 15 49 = 3×5 43×3
vr% ge ns[krs gSa fd mijksDr fu;eksa dk lR;kiu gqvk gSA
,d eq[; fcUnq: ,d dj.kh dh ?kkr dks cnyus ds fy, ge dj.kh dh ?kkr dks rFkk
dj.khxr dh ?kkr dks ,d gh /kukRed iw.kk±d ls xq.kk dj nsrs gSAa
mnkgj.k ds fy,
rFkk
3
2 = 6 22 = 6 4
4
3 = 8 32 = 8 9
2-12 ltkrh; djf.k;ka
nks djf.k;ka ltkrh; dgykrh gSa ;fn mUgsa ,d gh vifjes; xq.ku[kaM esa cnyk tk lds] pkgs
muds xq.kkad dqN Hkh gksAa
mnkgj.kr;k, 3 5 rFkk 7 5 ltkrh; djf.k;ka gSAa iqu% vki 75 = 5 3 rFkk 12 = 2 3
ij fopkj dhft,A ;gk¡ 75 rFkk 12 dks 5 3 rFkk 2 3 ds :i esa iznf'kZr fd;k x;k
gSA vr% og ltkrh; djf.k;ka gSAa
2-13 dj.kh dk ljyre ¼U;wure½ :i
,d dj.kh vius ljyre ¼U;wure½ :i esa gksrh gS ;fn blds
a) dj.kh fpUg dh U;wure laHko ?kkr gks
60
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
b) dj.kh fpUg ds varxZr dksbZ fHkUu u gks
c) n ?kkr ds dj.kh fpUg esa an, :i dk dksbZ xq.ku[akM u gks] tgk¡ a /ku iw.kk±d gS]
tSls,
3
125 3 125 ×12 5 3
=
=
12
18
18 ×12 6
fVIi.kh
vkb, dqN mnkgj.k ys%a
mnkgj.k 2.20: fuEu esa ls izR;sd dks 'kq) dj.kh ds ljyre :i esa O;ä dhft,
(i) 2 7
(ii) 44 7
(iii)
3
32
4
gy:
(i) 2 7 = 2 2 × 7 = 4 × 7 = 28 , tks ,d 'kq) dj.kh gSA
(ii) 44 7 = 4 4 4 × 7 = 4 256 × 7 = 4 1792 , tks ,d 'kq) dj.kh gSA
3
9
32 = 32 × = 18 , tks ,d 'kq) dj.kh gSA
4
16
(iii)
mnkgj.k 2.21: fuEu esa ls izR;sd dks fefJr dj.kh ds ljyre :i esa fyf[k,%
(i) 128
(ii) 6 320
(iii) 3 250
gy:
(i) 128 = 64 × 2 = 8 2 , tks ,d fefJr dj.kh gSA
(ii)
6
320 = 6 2 × 2 × 2 × 2 × 2 × 2 × 5
= 6 26 × 5 = 26 5 , tks ,d fefJr dj.kh gSA
(iii)
3
250 = 3 5 × 5 × 5 × 2 = 53 2 , tks ,d fefJr dj.kh gSA
ns[ksa vkius fdruk lh[kk 2-7
1. crkb, fuEu esa ls dkSu&dkSu ls ltkrh; dj.kh ;qXe gS%a
(i) 8 , 32
xf.kr
(ii) 5 3 ,6 18
(iii) 20 , 125
61
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
2. 'kq) dj.kh ds :i esa O;Dr dhft,%
(i) 7 3
(ii) 3 3 16
(iii)
5
24
8
3. fefJr dj.kh ds ljyre :i esa O;Dr dhft,%
fVIi.kh
(i) 3 250
(ii) 3 243
(iii) 4 512
2-14 djf.k;ksa ij pkjksa ewyHkwr lafØ;k,a
2.14.1 djf.k;ksa dk ;ksx rFkk O;odyu
ifjes; la[;kvksa dh rjg djf.k;ksa dk ;ksx rFkk O;odyu fd;k tkrk gS
5 3 + 17 3 = (5 + 17 ) 3 = 22 3
tSls,
rFkk
12 5 − 7 5 = [12 − 7] 5 = 5 5
djf.k;ksa dk ;ksx rFkk O;odyu djus ds fy, ge igys mUgsa ltkrh; djf.k;ka cukdj mu
ij lafØ;k,a djrs gSa
mnkgj.kr;k]
i) 50 + 288
=
5 × 5 × 2 + 12 × 12 × 2
= 5 2 + 12 2 = 2 (5 + 12 ) = 17 2
ii) 98 − 18
=
7 × 7 × 2 − 3× 3× 2
= 7 2 − 3 2 = (7 − 3) 2 = 4 2
mnkgj.k 2.22: fuEu esa ls izR;sd dks ljy dhft,%
(i) 4 6 + 2 54
(ii) 45 6 − 3 216
gy:
(i) 4 6 + 2 54
= 4 6 + 2 3× 3× 6
62
xf.kr
?kkrkad rFkk dj.kh
ekWM~;wy–1
chtxf.kr
= 4 6 + 6 6 = 10 6
(ii) 45 6 − 3 216
= 45 6 − 3 6 × 6 × 6
fVIi.kh
= 45 6 − 18 6
= 27 6
mnkgj.k 2.23: fn[kkb, fd
24 45 − 16 20 + 245 − 47 5 = 0
gy:
24 45 − 16 20 + 245 − 47 5
= 24 3 × 3 × 5 − 16 2 × 2 × 5 + 7 × 7 × 5 − 47 5
= 72 5 − 32 5 + 7 5 − 47 5
=
5 [72 − 32 + 7 − 47 ]
=
5 × 0 = 0 = RHS
mnkgj.k 2.24: ljy dhft,: 23 16000 + 83 128 − 33 54 + 4 32
gy:
23 16000 = 23 10 ×10 × 10 × 8 × 2 = 2 × 10 × 23 2 = 403 2
83 128 = 83 4 × 4 × 4 × 2 = 323 2
33 54 = 33 3 × 3 × 3 × 2 = 93 2
4
32 = 24 2
∴ okafNr O;atd
= 403 2 + 323 2 − 93 2 + 24 2
= (40 + 32 − 9)3 2 + 24 2
= 633 2 + 24 2
xf.kr
63
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
ns[ksa vkius fdruk lh[kk 2-8
fuEu esa ls izR;sd dks ljy dhft,%
fVIi.kh
1.
175 + 112
2.
32 + 200 + 128
3. 3 50 + 4 18
108 − 75
4.
5.
3
24 + 3 81 − 83 3
6. 63 54 − 23 16 + 43 128
7. 12 18 + 6 20 − 6 147 + 3 50 + 8 45
2.14.2 djf.k;ksa esa xq.kk rFkk Hkkx
nks djf.k;ksa esa xq.kk vFkok Hkkx fd;k tk ldrk gS ;fn mudh ,d gh dj.kh dh ?kkr gksA
geus lh[kk gS fd ,d dj.kh dh ?kkr cnyus ds fy, dj.kh dh ?kkr rFkk dj.khxr dh ?kkr
dks ,d gh /kukRed la[;k ls xq.kk vFkok Hkkx djuk gksrk gSA blfy, xq.kk vFkok Hkkx djus
ls igys ge mUgsa ,d gh ?kkr dh djf.k;ksa esa cnyrs gSAa
vkb, dqN mnkgj.k ysa
3 × 2 = 3× 2 = 6
3 rFkk
2 dh ?kkr leku gSa
]
12
= 6
2
12 ÷ 2 =
vkb, vc
[ 3 rFkk
3
2 dks xq.kk djsAa
3 = 6 33 = 6 27
3
2 =6 4
∴ 3 × 3 2 = 6 27 × 6 4 = 6 108
rFkk
3
3 6 27 6 27
= 6
=
4
2
4
vkb, dqN mnkgj.k ysAa
64
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
mnkgj.k 2.25: (i) 53 16 rFkk 113 40 dh xq.kk dhft,A
(ii) 153 13 dks 66 5 ls Hkkx dhft,A
gy:
(i) 53 16 × 113 40
fVIi.kh
= 5 ×11× 3 2 × 2 × 2 × 2 × 3 2 × 2 × 2 × 5
= 55 × 2 × 23 2
3
5
= 220 3 10
153 13 5 6 132 5 6 169
= . 6
=
(ii) 6
2 5
2
5
6 5
mnkgj.k 2.26: ljy dhft, rFkk ifj.kke dks ljyre :i esa O;Dr dhft,%
2 50 × 32 × 2 72
gy:
2 50 = 2 5 × 5 × 2 = 10 2
32 = 2 × 2 × 2 × 2 × 2 = 4 2
2 72 = 2 × 6 2 = 12 2
∴ fn;k gqvk O;atd = 10 2 × 4 2 × 12 2
= 960 2
2-15 djf.k;ksa dh rqyuk
djf.k;ksa dh rqyuk djus ds fy, ge igys mUgas ,d gh ?kkr dh djf.k;ksa esa cnyrs gSa vkSj
fQj muds dj.khxrksa dh muds xq.kkdksa lfgr rqyuk djrs gSAa vkb, dqN mnkgj.k ysAa
mnkgj.k 2.27:
1
rFkk
4
1
esa ls dkSu lk cM+k gS\
3
gy:
1 6 ⎛1⎞
1
= ⎜ ⎟ =6
4
64
⎝4⎠
3
3
3
xf.kr
1 6 1
=
3
9
65
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
1 1
1
1
1
1
>
⇒6 >6
⇒3 >
9 64
9
64
3
4
mnkgj.k 2.28: vkjksgh Øe esa O;ofLFkr dhft,:
3
2,
3 rFkk
6
5.
fVIi.kh
gy:
2, 3, rFkk 6 dk y- l- 6 gSA
∴ 3 2 = 6 22 = 6 4
3 = 6 33 = 6 27
6
vc
6
5=6 5
4 < 6 5 < 6 27
⇒3 2<6 5< 3
ns[ksa vkius fdruk lh[kk 2-9
1.
3
3 rFkk 3 5 dks ijLij xq.kk dhft,A
2.
3.
32 rFkk 53 4 dks ijLij xq.kk dhft,A
3
135 dks 3 5 ls Hkkx nhft,A
4. 2 24 dks 3 320 ls Hkkx nhft,A
5. dkSu lk cM+k gS%
6. dkSu lk NksVk gS%
4
5 or 3 4 ?
5
10 or 4 9 ?
7. vkjksgh Øe eas O;ofLFkr dhft,%
3
2,
6
3,
3
4
8. vojksgh Øe eas O;ofLFkr dhft,%
3
66
2,
4
3,
3
4
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
2-16 djf.k;ksa dk ifjes;dj.k
fuEu xq.kuQyksa ij fopkj dhft,%
1
1
(i) 3 2 × 3 2 = 3
7
fVIi.kh
4
(ii) 511 × 511 = 5
1
3
(iii) 7 4 × 7 4 = 7
mijksDr rhu xq.kuQyksa esa ls izR;sd esa ge ns[krs gSa fd nks djf.k;ksa dks xq.kk djus ij gesa
ifj.kke ,d ifjes; la[;k feyrh gSA ,slh n'kk esa izR;sd dj.kh dks nwljh dj.kh dk
ifjes;dkjh xq.kd dgrs gSAa
(i)
3 dk ifjes;dkjh xq.kd
(ii)
11
54 dk ifjes;dkjh xq.kd
(iii)
4
7 dk ifjes;dkjh xq.kd
3 gS rFkk foykser%
11
4
57 gS rFkk foykser%
73 gS
rFkk foykser%
nwljs 'kCnksa esa djf.k;ksa dks ifjes; la[;kvksa esa cnyus dh izfØ;k dks ge ifjes;dj.k dgrs
gSa rFkk os nks la[;k,a] tks xq.kk djus ij ifj.kke ,d ifjes; la[;k nsrh gS]a ,d nwljs dk
ifjes;dkjh xq.kd dgh tkrh gSAa
mnkgj.kr;k%
x dk ifjes;dkjh xq.kd
x gS]
3 + 2 dk
3 − 2 gSA
uksV:
(i) x − y rFkk x + y la;qXeh djf.k;ka dgykrh gSAa
(ii) ifjes;dj.k dk iz;ksx izk;% fHkUu ds gj dk ifjes;dj.k djus ds fy, fd;k tkrk gSA
vkb, dqN mnkgj.k ys%a
mnkgj.k 2.29: 18 rFkk 12 ds ifjes;dkjh xq.kd Kkr dhft,A
gy:
18 =
3× 3× 2 = 3 2
vr% ifjes;dkjh xq.kd
12 =
2 gSA
2× 2×3 = 2 3 .
vr% ifjes;dkjh xq.kd 3 gSA
xf.kr
67
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
fVIi.kh
mnkgj.k 2.30:
2+ 5
ds gj dk ifjes;dj.k dhft,A
2− 5
gy:
2+ 5
=
2− 5
gy:
(
(
)(
)(
4+3 5 4+3 5
4+3 5
=
4−3 5 4+3 5
4−3 5
mnkgj.k 2.32:
) (
)
2+ 5
=
2+ 5
2+ 5
−3
)
2
)
)
16 + 45 + 24 5
61 24
=− −
5
16 − 45
29 29
1
3 − 2 +1
1
3 − 2 +1
ds gj dk ifjes;dj.k dhft,A
=
(
(
=
3 − 2 −1
)
2
3 − 2 −1
=
3 − 2 −1
4−2 6
3 − 2 −1 4 + 2 6
×
4−2 6
4+2 6
=
4 3 −4 2 −4+6 2 −4 3 −2 6
16 − 24
=−
mnkgj.k 2.33: ;fn
)
3 − 2 −1
[( 3 − 2 )+1] [( 3 − 2 )−1]
=
68
)(
)(
4+3 5
ds gj dk ifjes;dj.k dhft,A
4−3 5
=
gy:
2+ 5
2− 5
7 + 2 10
7 2
=− −
10
3
3 3
=−
mnkgj.k 2.31:
(
(
2 −2− 6
6− 2+2
=
4
4
3+ 2 2
= a + b 2 , gS] rks a rFkk b ds eku Kkr dhft,A
3− 2
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
3+ 2 2 3+ 2 2 3+ 2 9+ 4+9 2
=
×
=
9−2
3− 2
3− 2 3+ 2
gy:
=
13 + 9 2 13 9
= +
2 =a+b 2
7
7 7
vr% a =
13
,
7
b=
fVIi.kh
9
7
ns[ksa vkius fdruk lh[kk 2-10
1. fuEu esa ls izR;sd dk ifjes;dkjh xq.kd Kkr dhft,%
(iii) 3 x 2 + 3 y 2 + 3 xy
(ii) 2 + 1
(i) 3 49
2. fuEu esa ls izR;sd ds gj dk ifjes;dj.k djds ljy dhft,%
(i)
12
5
(ii)
3. ljy dhft,:
4.
2 3
17
(iii)
11 − 5
11 + 5
(iv)
3 +1
3 −1
2+ 3 2− 3
+
2− 3 2+ 3
1
ds gj dk ifjes;dj.k dhft,A
3 − 2 −1
1
5. ;fn a = 3 + 2 2 gS rks] a + dk eku Kkr dhft,A
a
6. ;fn
2+5 7
= x + 7 y gS] rks x rFkk y ds eku Kkr dhft,A
2−5 7
vkb, nksgjk,¡
•
a × a × a × ..... m ckj = am ?kkrkadh; :i gS tgk¡ a vk/kkj gS rFkk m ?kkrkad gSA
•
?kkrkadksa ds fu;e gSa%
m
(i) a × a = a
m
xf.kr
n
m+n
(ii) a ÷ a = a
m
n
m–n
(iii) (ab) = a b
m
m m
am
⎛a⎞
(iv) ⎜ ⎟ = m
b
⎝b⎠
69
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
( )
n
(v) a m
fVIi.kh
−m
=
1
am
p
•
aq = ap
•
,d vifjes; la[;k n x dj.kh dgykrh gS] ;fn x ,d ifjes; la[;k gS rFkk n x esa]
n dj.kh dh ?kkr gS rFkk x dj.khxr dgykrk gSA
•
,d dj.kh ftlesa xq.ku[kaM 1 ds vfrfjDr dksbZ vkSj la[;k gks] ,d fefJr dj.kh
dgykrh gSA
•
dj.kh dh ?kkr og la[;k gS tks ewy dks n'kkZrh gSA
•
•
q
n
x
•
dh ?kkr n gSA
dj.kh fpUg ds fu;e (a > 0, b > 0)
n
[ a] = a
n
(i)
n
(ii) a × b = ab
n
n
n
(iii)
n
a n a
=
b
b
djf.k;ksa ij lafØ;k,a
⎛
x × y = (xy ) ; ⎜⎜ x
⎝
1
n
(x )
1
m n
70
(vii) a
(vi) ao = 1
= a mn
1
n
1
n
m
n
=x ;
m
a
x =
1
n
1
m
⎞
⎟ =x
⎟
⎠
mn
x
an
1
mn
⎛
= ⎜⎜ x
⎝
( )
or x
1
a m
1
m
1
n
1
n
1
⎛ x ⎞n
x
⎞
⎟ ; 1 = ⎜⎜ y ⎟⎟
⎟
yn ⎝ ⎠
⎠
a
m
=x =x
an
mn
( )
= x an
1
mn
•
djf.k;ka ltkrh; gksrh gSa ;fn muds vifjes; xq.ku[kaM leku gksAa
•
ltkrh; djf.k;ksa dk ;ksx rFkk O;odyu fd;k tk ldrk gSA
•
djf.k;ksa dh ?kkr dks cnyk tk ldrk gSA blds fy, dj.kh dh ?kkr rFkk dj.khxr
dh ?kkr dks ,d gh /kukRed la[;k ls xq.kk fd;k tkrk gSA
•
ltkrh; djf.k;ksa ds ijLij xq.kk rFkk Hkkx fd;k tk ldrk gSA
•
djf.k;ksa dh rqyuk djus ds fy, ge djf.k;ksa dks ,d gh ?kkr okyh djf.k;ksa esa cny
ysrs gSa vkSj fQj muds dj.khxrksa ¼xq.kkadksa lfgr½ ls mudh rqyuk djrs gSAa
•
;fn nks djf.k;ksa dk xq.ku ,d ifjes; la[;k gS] rks izR;sd dks nwljs dk ifjes;dkjh
xq.kd dgk tkrk gSA
•
x + y , dks x − y dk ifjes;dkjh xq.kd dgk tkrk gS rFkk foykser%
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
vkb, vH;kl djsa
1. ?kkrkadh; :i esa O;Dr dhft,%
fVIi.kh
(i) 5 × 3 × 5 × 3 × 7 × 7 × 7 × 9 × 9
⎛−7⎞ ⎛ −7⎞ ⎛ −7⎞ ⎛ −7⎞
⎟×⎜
⎟×⎜
⎟×⎜
⎟
(ii) ⎜
⎝ 9 ⎠ ⎝ 9 ⎠ ⎝ 9 ⎠ ⎝ 9 ⎠
2. ljy dhft,%
3
2
⎛ 5⎞ ⎛7⎞ ⎛3⎞
(i) ⎜ − ⎟ × ⎜ ⎟ × ⎜ ⎟
⎝ 6⎠ ⎝5⎠ ⎝7⎠
2
⎛ 3 ⎞ 35 ⎛ 1 ⎞
(ii) ⎜ ⎟ × × ⎜ − ⎟
⎝ 7 ⎠ 27 ⎝ 5 ⎠
3
2
3. ljy dhft, rFkk ifj.kke dks ?kkrkadh; :i esa O;Dr dhft,%
(i) (10 )2 × (6 )2 × (5)2
20
⎛ 37 ⎞
⎛ 37 ⎞
(ii) ⎜ − ⎟ ÷ ⎜ − ⎟
⎝ 19 ⎠
⎝ 19 ⎠
⎡⎛ 3 ⎞3 ⎤
(iii) ⎢⎜ ⎟ ⎥
⎢⎣⎝ 13 ⎠ ⎥⎦
20
5
4. ljy dhft,%
(i) 3o + 7o + 37o – 3
(ii) ( 7o + 3o) ( 7o – 3o)
5. ljy dhft,%
−3
(i) (32 ) ÷ (32 )
12
−6
⎛ 2⎞
⎛ 2⎞
(iii) ⎜ − ⎟ × ⎜ − ⎟
⎝ 9⎠
⎝ 9⎠
(ii) (111) × (111)
−5
6
−3
11
⎛3⎞
⎛3⎞
⎛3⎞
6. x dk eku Kkr dhft, ;fn ⎜ ⎟ × ⎜ ⎟ = ⎜ ⎟
⎝7⎠
⎝7⎠
⎝7⎠
−2
−9
x
⎛3⎞
⎛3⎞
⎛3⎞
7. x dk eku Kkr dhft, ;fn ⎜ ⎟ × ⎜ ⎟ = ⎜ ⎟
⎝ 13 ⎠
⎝ 13 ⎠
⎝ 13 ⎠
xf.kr
5
2 x +1
71
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
8. fuEu esa ls izR;sd dks vHkkT; xq.ku[kaMksa ds xq.ku ds :i esa Kkr dj ifj.kke dks
?kkrkadh; :i esa O;Dr dhft,
(i) 6480000
fVIi.kh
(ii) 172872
(iii) 11863800
9. lkbjl flrkjk i`Fohls 8.1 × 1013 fdeh dh nwjh ij gSA ;g ekurs gq, fd izdk'k 3.0
× 105 fdeh dh xfr ls ;k=k djrk gS] rks Kkr dhft, fd lkbjl ls izdk'k fdruh
nsj esa i`Foh rd igqp
a sxk\
10. fuEu esa ls dkSu&dkSu lh djf.k;ka gS\a
(i)
36
289
(iii) 3
(ii) 9 729
5 +1
(iv) 4 3125
11. 'kq) dj.kh ds :i esa O;Dr dhft,%
(ii) 53 4
(i) 32 3
(iii) 55 2
12. fuEu dks ljyre :i dh fefJr dj.kh ds :i esa O;Dr dhft,%
(i) 4 405
(ii) 5 320
(iii) 3 128
13. fuEu esa ls dkSu ls ltkrh; dj.kh ds ;qXe gS%a
(i) 112 , 343
(ii) 3 625 , 3 3125 × 25
(iii) 6 216 , 250
14. fuEu esa ls izR;sd dks ljy dhft,%
(i) 4 48 −
5 1
+6 3
2 3
(ii) 63 + 28 − 175
(iii)
8 + 128 − 50
15. dkSu lk cM+k gS
(i) 2 vFkok 3 3 ?
(ii) 3 6 vFkok 4 8 ?
16. vojksgh Øe esa yxkb,%
(i) 3 , 3 4 , 4 5
(ii) 2 , 3 , 3 4
17. vkjksgh Øe esa yxkb,%
3
72
16 , 12 , 6 320
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
18. gj dk ifjes;dj.k djds ljy dhft,%
3
6− 7
(i)
(ii)
12
7− 3
(iii)
5 −2
5+2
19. fuEu esa ls izR;sd ds gj dk ifjes;dj.k djds ljy dhft,%
(i)
1
1+ 2 − 3
(ii)
fVIi.kh
1
7 + 5 − 12
5+2 3
= a + b 3, gS rks] a rFkk b ds eku Kkr dhft, tgk¡ a rFkk b ifjes;
7+4 3
20. ;fn
la[;k,a gSA
1
21. ;fn x = 7 + 4 3 gS rks] x + dk eku Kkr dhft,A
x
ns[ksa vkius fdruk lh[kk ds mÙkj
2.1
10
1. (i) (–7)
⎛3⎞
(ii) ⎜ ⎟
⎝4⎠
2.
?kkr
4
vk/kkj
(i) – 3
5
(ii) 7
4
(iii) −
2
11
81
2401
(ii)
16
6561
4. (i)
3
7
(ii)
625
324
xf.kr
5
20
8
3. (i)
⎛1⎞
5. (i) ⎜ ⎟
⎝3⎠
⎛ −5⎞
⎟
(iii) ⎜
⎝ 7 ⎠
⎛ 1⎞
(ii) ⎜ − ⎟
⎝ 7⎠
(iii) −
4
27
64
⎛ 5⎞
(iii) ⎜ − ⎟
⎝ 3⎠
4
73
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
2.2
1. (i) 31 × 111 × 131
(ii) 23 × 34
(iii) 23 × 33 × 71
2. (i) 36
(ii) 29
(iii) 25 × 34
fVIi.kh
⎛ 3⎞
(v) ⎜ − ⎟
⎝ 2⎠
113
(iv) 12
2
5
2.3
5
1. (i) (7)
2
2. (i) (–7)
3. (i) 2
18
⎛ 11 ⎞
(iv) ⎜ ⎟
⎝3⎠
5
⎛3⎞
(ii) ⎜ ⎟
⎝4⎠
5
⎛ 7⎞
(iii) ⎜ − ⎟
⎝ 8⎠
⎛3⎞
(ii) ⎜ ⎟
⎝4⎠
6
⎛ 7⎞
(iii) ⎜ − ⎟
⎝ 3⎠
⎛3⎞
(ii) ⎜ ⎟
⎝4⎠
6
⎛ 5⎞
(iii) ⎜ − ⎟
⎝ 9⎠
⎛ 7⎞
(v) ⎜ − ⎟
⎝ 11 ⎠
6
15
15
3
4. lgh: (i), (ii), (vii)
xyr: (iii), (iv), (v), (vi)
2.4
1.
49
9
⎛7⎞
2. (i) ⎜ ⎟
⎝3⎠
4
⎛7⎞
3. (i) ⎜ ⎟
⎝3⎠
−4
4. (i)
81
16
12
⎛ 13 ⎞
(iii) ⎜ ⎟
⎝3⎠
2
(ii) 12
⎛1⎞
(ii) ⎜ ⎟
⎝7⎠
(ii) −
2
3
−10
⎛ 4⎞
(iii) ⎜ − ⎟
⎝ 3⎠
(iii) −
−10
343
125
5. lgh: (ii), (iii), (iv), xyr: (i), (v)
74
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
2.5
1. (i) 8
(ii)
25
9
2. (i) 1
(ii)
7
8
(iii)
13
16
fVIi.kh
2.6
1. (i) 4, 6, 2
(ii) 64, 343, 119
2. (iii), (iv)
3. 'kq) dj.kh: (i), (iv)
fefJr dj.kh: (ii), (iii)
2.7
1. (i), (iii)
75
8
2. (i) 147
(ii) 3 432
(iii)
3. (i) 53 2
(ii) 33 9
(iii) 44 2
1. 9 7
2. 22 2
3. 27 2
5. − 33 3
6. 303 2
7. 51 2 + 36 5 − 42 3
2.8
4.
3
2.9
1. 203 2
2. 33 5
3. 3
5.
6.
7.
3
4
4
9
6
3, 3 2 , 3 4
4. 6
216
25
8.
4 , 4 3, 3 2
3
2.10
1. (i) 3 7
2. (i)
xf.kr
12
5
5
(ii) 2 − 1
(ii)
2 51
17
(iii) 3 x − 3 y
(iii)
8
55
−
3
3
(iv) 2 + 3
75
ekWM~;wy–1
xf.kr ek/;fed ikB~;Øe
chtxf.kr
3. 14
4. −
fVIi.kh
[
1
2+ 6 + 2
4
]
5. 6
6. −
179 20 7
−
171 171
vkb, vH;kl djsa ds mÙkj
2
2
3
2
1. (i) 5 × 3 × 7 × 9
2. (i) −
5
56
⎛ 7⎞
(ii) ⎜ − ⎟
⎝ 9⎠
(ii)
4
1
105
15
4
2
3. (i) 2 × 3 × 5
4. (i) 'kwU;
4
(ii) 1
⎛3⎞
(iii) ⎜ ⎟
⎝ 13 ⎠
(ii) 'kwU;
18
5. (i) (32)
(ii) 111
⎛2⎞
(iii) ⎜ ⎟
⎝9⎠
2
6. x = 8
7. x = – 6
8. 27 × 34 × 54
9. 33 × 107 ls10. (ii), (iii), (iv)
11. (i) 2 27
(ii) 3 500
(iii) 5 6250
12. (i) 34 5
(ii) 25 10
(iii) 43 2
13. (i), (ii)
76
xf.kr
ekWM~;wy–1
?kkrkad rFkk dj.kh
chtxf.kr
14. (i)
127
3
6
(ii) 'kwU;
(iii) 5 2
15. (i) 3 3
(ii) 3 6
16. (i) 3 , 3 4 , 4 5
(ii) 3 , 3 4 , 2
fVIi.kh
17. 3 16 , 6 320 , 12
(
18. (i) − 3 6 + 7
19. (i)
2+ 2 + 6
4
)
(
(ii) 3 7 + 3
(ii)
)
(ii) 9 − 4 5
7 5 + 5 7 + 2 105
70
20. a = 11, b = –6
21. 14
xf.kr
77